Symmetries in collinear effective theory

We consider various symmetries present in the collinear effective theory and their implications. There are collinear, soft and ultrasoft gauge symmetries and we discuss transformation properties of a collinear quark and gauge fields under these gauge transformations. If we require the gauge invariance order by order, the gauge fields have definite transformation properties. We also probe reparameterization invariance and residual energy invariance in the collinear effective theory and discuss their physical consequences. When we describe strong interaction processes which include massless particles with large energy E, we can use the collinear effective theory [1–4]. In the collinear effective theory, the effective Lagrangian can be systematically expanded in powers of λ ∼ p⊥/E where p⊥ is the transverse momentum of a collinear quark. Even though there is one large parameter E, there are three scales involved in the effective theory. The momentum of an energetic particle can be written as p = 1 2 (n · p)n + pμ⊥ + 1 2 (n · p)n, where n · p ∼ λ, p⊥ ∼ λ, and n · p ∼ λ. If we include small fluctuations due to the strong interaction, the momentum P μ of an energetic particle is decomposed as the sum of a large momentum p with n · p ∼ λ, pμ⊥ ∼ λ and n · p = 0 and a small residual momentum k ∼ λ: P μ = p + k = n 2 n · p+ pμ⊥ + k . (1) Here p acts as a label momentum on collinear fields, and the residual momentum k is a small fluctuation. Since there are three distinct scales E, Eλ, Eλ, we have to construct effective theories at each scale. Here we consider an effective theory in the region Eλ < 1 e-mail address: [email protected] Preprint submitted to Elsevier Preprint 1 February 2008 μ < E, in which collinear degrees of freedom above μ are integrated out. This effective theory can be used in exclusive decays of a heavy meson into energetic light mesons. It has been applied to the resummation of Sudakov logarithms in B decays [1] and to the proof of factorization in B → Dπ [5]. And the calculation of the form factors in heavy-to-light transitions to order λ was performed [6]. The symmetry structure in the collinear effective theory is rich. There are interesting physical implications by combining the symmetries of the effective theory. In this Letter, we consider gauge invariance, reparameterization invariance and residual energy invariance. In order to discuss gauge symmetries of the collinear effective theory, let us consider the full QCD first. The Lagrangian for a massless quark and gluons is given by LQCD = qi/ Dq − 1 4 GaμνG , (2) where the covariant derivative D is defined as Dμ = ∂μ + igA a μT a and Gμν is the gluon field strength tensor. The quark field and the gauge fields transform under an arbitrary SU(3) gauge transformation of the form U = e a(x)Ta as q(x) → U(x)q(x), A(x) → U(x)A(x)U (x)− i g U(x)∂U (x), (3) and the Lagrangian is invariant under the gauge transformation. In the collinear effective theory, we decompose each field into a collinear field, a soft field and an ultrasoft (usoft) field. For a collinear quark, we redefine the field by extracting large momenta as qn,p(x) = ∑